Why can't any way of rounding seats be perfectly fair?

Because seats come in whole numbers, and fair shares do not. Mathematically, an apportionment method is just a method of rounding real numbers to natural numbers. Despite the simplicity of this problem, every method of rounding suffers one or more paradoxes, as proven by the Balinski–Young theorem.

Certain quantities, like milk, can be divided in any proportion whatsoever; others, such as horses, cannot, and only whole numbers will do. In the latter case, there is an inherent tension between the desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values.

The tension surfaced in United States congressional apportionment. After the 1880 census, C. W. Seaton, chief clerk of the Census Bureau, computed apportionments for all House sizes between 275 and 350 and discovered that Alabama would get eight seats with a House size of 299 but only seven with a House size of 300. In general, the Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares.

In 1982, two mathematicians, Michel Balinski and Peyton Young, proved that any method of apportionment that does not violate the quota rule will result in paradoxes whenever there are four or more parties (or states, regions, etc.). The quota rule says each party gets one of the two whole numbers closest to its fair share. For example, if a party's fair share is 7.34 seats, it must get either 7 or 8 seats to avoid a violation; any other number will violate the rule.